A really simple elementary proof of the uniform boundedness theorem

TL;DR

This paper presents an elementary and extremely simple proof of the uniform boundedness theorem that avoids using the Baire category theorem, making the proof more accessible.

Contribution

It introduces a novel, elementary proof of the uniform boundedness theorem that simplifies the existing complex proofs.

Findings

Proof is elementary and does not rely on Baire category theorem

Simplifies understanding of the uniform boundedness theorem

Accessible to a broader audience

Abstract

I give a proof of the uniform boundedness theorem that is elementary (i.e. does not use any version of the Baire category theorem) and also extremely simple.